**Astérisque**

Volume: 343;
2012;
169 pp;
Softcover

MSC: Primary 55; 14;
**Print ISBN: 978-2-85629-342-3
Product Code: AST/343**

List Price: $60.00

AMS Member Price: $48.00

# String Topology for Stacks

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*Kai Behrend; Grégory Ginot; Behrang Noohi; Ping Xu*

A publication of the Société Mathématique de France

The authors establish the general machinery of string topology for
differentiable stacks. This machinery allows them to treat on equal footing free
loops in stacks and hidden loops. They construct a bivariant (in the sense of
Fulton and MacPherson) theory for topological stacks: it gives them a flexible
theory of Gysin maps, which are automatically compatible with pullback,
pushforward and products. Then the authors prove an excess formula in this
context.

The authors introduce oriented stacks, generalizing oriented
manifolds, which are stacks on which they can do string topology. They
prove that the homology of the free loop stack of an oriented stack
and the homology of hidden loops (sometimes called ghost loops) are
Frobenius algebras which are related by a natural morphism of Frobenius
algebras. They also prove that the homology of the free loop stack has a
natural structure of \(BV\)-algebra which, together with the Frobenius
structure, fits into homological conformal field theories with
closed positive boundaries. They also use their constructions to study
an analogue of the loop product for stacks of maps of
(\(n\)-dimensional) spheres to oriented stacks and compatible
power maps in their homology.

Using their general machinery, the authors construct an
intersection pairing for (not necessarily compact) almost complex
orbifolds which is in the same relation to the intersection pairing
for manifolds as Chen-Ruan orbifold cup-product is to ordinary
cup-product of manifolds. They show that the hidden product of almost
complex orbifolds is isomorphic to the orbifold intersection pairing
twisted by a canonical class. Finally they gave some examples, including
the case of the classifying stacks \([*/G]\) of a compact Lie
group.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians interested in string topology, topological stacks, and loop stacks.